Combine water, 2 cups maple syrup, and salt in a large pot; stir until brine is evenly combined, about 2 minutes. Add pork butt to brine; refrigerate for 2 hours 45 minutes.
Soak wood chips in a bowl of warm water while pork is brining.
Drain wood chips and place in the smoker. Place grill rack on top. Remove pork from brine and place on grill rack in smoker, discarding brine.
Turn smoker temperature to 280 degrees F (138 degrees C). Combine apple cider vinegar and 1/4 cup maple syrup in a clean spray bottle.
Cook pork in the smoker, spraying every 45 minutes with vinegar spray, until internal temperature of pork reaches 195 degrees F (90 degrees C) to 200 degrees F (93 degrees C), about 12 hours. Transfer pork to a cutting board and slice or shred.